Simplify the following expression and state the condition under which the simplification is valid. You can assume that $a \neq 0$. $y = \dfrac{-7}{12a - 42} \div \dfrac{a}{2a(2a - 7)} $
Answer: Dividing by an expression is the same as multiplying by its inverse. $y = \dfrac{-7}{12a - 42} \times \dfrac{2a(2a - 7)}{a} $ When multiplying fractions, we multiply the numerators and the denominators. $y = \dfrac{ -7 \times 2a(2a - 7) } { (12a - 42) \times a } $ $ y = \dfrac {-7 \times 2a(2a - 7)} {a \times 6(2a - 7)} $ $ y = \dfrac{-14a(2a - 7)}{6a(2a - 7)} $ We can cancel the $2a - 7$ so long as $2a - 7 \neq 0$ Therefore $a \neq \dfrac{7}{2}$ $y = \dfrac{-14a \cancel{(2a - 7})}{6a \cancel{(2a - 7)}} = -\dfrac{14a}{6a} = -\dfrac{7}{3} $